2000 Fiscal Year Final Research Report Summary
DEVELOPEMENT OF RENORMALIZATION GROUP METHODS AS TOOLS OF ANALYSIS AND THEIR APPLICATIONS TO DYNAMICAL SYSTEMS
| Project/Area Number |
11640220
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | SETSUNAN UNIVERSITY |
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Principal Investigator |
ITO Keiichi r. SESTUNSN UNIVERSITY, MATHEMATICS DEPARTMENT, PROFESSOR, 工学部, 教授 (50268489)
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| Co-Investigator(Kenkyū-buntansha) |
TERAMOTO Yoshiaki SESTUNSN UNIVERSITY, MATHEMATICS DEPARTMENT, ASSOCIATE PROFESSOR, 工学部, 助教授 (40237011)
ONO Hiroaki SESTUNSN UNIVERSITY, PHYSICS DEPARTMENT, PROFESSOR, 工学部, 教授 (50100780)
IKEBE Teruo SESTUNSN UNIVERSITY, MATHEMATICS DEPARTMENT, PROFESSOR, 工学部, 教授 (00025280)
WATARAI Seizo SESTUNSN UNIVERSITY, MATHEMATICS DEPARTMENT, ASSOCIATE PROFESSOR, 工学部, 助教授 (20131500)
SHIMADA Shin-ichi SESTUNSN UNIVERSITY, MATHEMATICS DEPARTMENT, ASSOCIATE PROFESSOR, 工学部, 助教授 (40196481)
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| Project Period (FY) |
1999 – 2000
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| Keywords | Renormalization Group / Recursion Formula / Scaling / O(N)Spin Model / Pauli-Fierz Model / Navier-Stokes Equation / Couette Flow / Hopf Bifurcation / Kolmogorov law |
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| Research Abstract |
1. Ito and Tamura (Kanazawa Univ.) studied classical O(N) symmetric spin model by renormalization group (block spin transformation) method. In the first stage, they argued the integrability of the functional determinent det^<N/2>(1+2iG_ψ/√<N>) with respect to ψ, where ψ is the auxially field introcuced for Fourier Transformation. Using the technique called polymer (cluster) expansion, they showed that the inverse critical temperature β_c obeys the bound β_c>N log N in two dimensions, which implies the existence of strong deviation. (β_c〜N for the dimension more than or equal to 3.) It is believed that β_c=∞ in the present model. To establish this conjecture, they recursively apply the BST to the model to decompose the determinant into product of many determinants which comes from fluctuations of various distance scales. They showed that the main part of the recursion relations is quite simple, and reproduces the flow of the hiererchical approximation of Wilson-Dyson type. It remains to
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control small non-local terms of the recursion formulas to solve the problem completely.
2. Ito and Hiroshima (Setsunan Univ.) investigate the Pauli-Fierz Model which is rgarded as a classical Quantum Electrodynamics (QED). Though QED is believed to be trivial if no momentum cutoff is introduced, the Pauli-Fierz model may not. They apply the renormalization group type argumemt to the Pauli-Fierz model (this idea is originally due to J.Froelich(ETH)). But their analysis remains to be seen.
3. Teramoto considere Couette-Taylor problems of the perturbation to the Couette flow between two rotating cylinders, and shows that the stationary bifurcation or Hopf bifurcation occurs when the Taylor number increases. He proved it with the help of numerical analysis by computer.
4. Teramoto and Ito investigated properties of turbulence, among them, the Kolmogorov law about the dissipation of energy and deviation from it. They tried to derive the deviation from the Navier-Stokes equation but they could not obtain concrete results this year.
5. Shimada characterized all possible selfadjoint extensions of Aharonov-Bohm hamiltonian in terms of boundary conditions at origin which distinguish the operator treated by Aharonov and Bohm from other possible ones. He developed scattering theory for their operators and obtained the phase shift formula. Less
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